How does batch size affect model size?
The model size (i.e., the number of parameters) is completely independent of the batch size you use during the training. 

Whatever the batch size is, your model processes each input within the batch independently. Concretely, batch size affects how many samples your model has to process before it makes an update. 

For example, when batch size is 1, it computes the gradients and updates the parameters for each input. When batch size is the whole dataset, it computes the gradients for each input in the dataset independently. It then takes the average of those gradients and uses this average to update the model, so there will be a single update per epoch. 

In your case, the model takes an input of size (width=224, height=224, color_channels=3) whatever batch size you use during the training.